The $\mathbb{F}_2$-Rank and Size of Graphs

Gal Beniamini; Asaf Etgar; Yael Kirkpatrick

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arXiv · 2206.11625

The $\mathbb{F}_2$-Rank and Size of Graphs

Gal Beniamini·Asaf Etgar·Yael Kirkpatrick

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Abstract

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we show that this is an infinite family of strongly-regular quasi-random graphs whose signed adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a combinatorial proof that for all odd $n$, no twin-free graphs of minimal $\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is attainable, which is tight.

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Gal Beniamini, Asaf Etgar, Yael Kirkpatrick. 2022-09-19. The $\mathbb{F}_2$-Rank and Size of Graphs. https://arxiv.org/abs/2206.11625

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